Structure ringNormTheory


Source File Identifier index Theory binding index

signature ringNormTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val interp_p_def : thm
    val polynom_TY_DEF : thm
    val polynom_case_def : thm
    val polynom_normalize_def : thm
    val polynom_simplify_def : thm
    val polynom_size_def : thm
    val r_canonical_sum_merge_def : thm
    val r_canonical_sum_prod_def : thm
    val r_canonical_sum_scalar2_def : thm
    val r_canonical_sum_scalar3_def : thm
    val r_canonical_sum_scalar_def : thm
    val r_canonical_sum_simplify_def : thm
    val r_ics_aux_def : thm
    val r_interp_cs_def : thm
    val r_interp_m_def : thm
    val r_interp_sp_def : thm
    val r_interp_vl_def : thm
    val r_ivl_aux_def : thm
    val r_monom_insert_def : thm
    val r_spolynom_normalize_def : thm
    val r_spolynom_simplify_def : thm
    val r_varlist_insert_def : thm
  
  (*  Theorems  *)
    val canonical_sum_merge_def : thm
    val canonical_sum_prod_def : thm
    val canonical_sum_scalar2_def : thm
    val canonical_sum_scalar3_def : thm
    val canonical_sum_scalar_def : thm
    val canonical_sum_simplify_def : thm
    val datatype_polynom : thm
    val ics_aux_def : thm
    val interp_cs_def : thm
    val interp_m_def : thm
    val interp_sp_def : thm
    val interp_vl_def : thm
    val ivl_aux_def : thm
    val monom_insert_def : thm
    val polynom_11 : thm
    val polynom_Axiom : thm
    val polynom_case_cong : thm
    val polynom_case_eq : thm
    val polynom_distinct : thm
    val polynom_induction : thm
    val polynom_nchotomy : thm
    val polynom_normalize_ok : thm
    val polynom_simplify_ok : thm
    val spolynom_normalize_def : thm
    val spolynom_simplify_def : thm
    val varlist_insert_def : thm
  
  val ringNorm_grammars : type_grammar.grammar * term_grammar.grammar
  
  val IMPORT : abstraction.inst_infos ->
    { interp_p_def : thm,
  polynom_case_def : thm,
  polynom_normalize_def : thm,
  polynom_simplify_def : thm,
  polynom_size_def : thm,
  polynom_TY_DEF : thm,
  r_canonical_sum_merge_def : thm,
  r_canonical_sum_prod_def : thm,
  r_canonical_sum_scalar2_def : thm,
  r_canonical_sum_scalar3_def : thm,
  r_canonical_sum_scalar_def : thm,
  r_canonical_sum_simplify_def : thm,
  r_ics_aux_def : thm,
  r_interp_cs_def : thm,
  r_interp_m_def : thm,
  r_interp_sp_def : thm,
  r_interp_vl_def : thm,
  r_ivl_aux_def : thm,
  r_monom_insert_def : thm,
  r_spolynom_normalize_def : thm,
  r_spolynom_simplify_def : thm,
  r_varlist_insert_def : thm,
  canonical_sum_merge_def : thm,
  canonical_sum_prod_def : thm,
  canonical_sum_scalar2_def : thm,
  canonical_sum_scalar3_def : thm,
  canonical_sum_scalar_def : thm,
  canonical_sum_simplify_def : thm,
  datatype_polynom : thm,
  ics_aux_def : thm,
  interp_cs_def : thm,
  interp_m_def : thm,
  interp_sp_def : thm,
  interp_vl_def : thm,
  ivl_aux_def : thm,
  monom_insert_def : thm,
  polynom_11 : thm,
  polynom_Axiom : thm,
  polynom_case_cong : thm,
  polynom_case_eq : thm,
  polynom_distinct : thm,
  polynom_induction : thm,
  polynom_nchotomy : thm,
  polynom_normalize_ok : thm,
  polynom_simplify_ok : thm,
  spolynom_normalize_def : thm,
  spolynom_simplify_def : thm,
  varlist_insert_def : thm }
  
(*
   [canonical] Parent theory of "ringNorm"
   
   [ring] Parent theory of "ringNorm"
   
   [interp_p_def]  Definition
      
      ⊢ (∀r vm c. interp_p r vm (Pconst c) = c) ∧
        (∀r vm i. interp_p r vm (Pvar i) = varmap_find i vm) ∧
        (∀r vm p1 p2.
           interp_p r vm (Pplus p1 p2) =
           r.RP (interp_p r vm p1) (interp_p r vm p2)) ∧
        (∀r vm p1 p2.
           interp_p r vm (Pmult p1 p2) =
           r.RM (interp_p r vm p1) (interp_p r vm p2)) ∧
        ∀r vm p1. interp_p r vm (Popp p1) = r.RN (interp_p r vm p1)
   
   [polynom_TY_DEF]  Definition
      
      ⊢ ∃rep.
          TYPE_DEFINITION
            (λa0'.
                 ∀ $var$('polynom').
                   (∀a0'.
                      (∃a. a0' =
                           (λa.
                                ind_type$CONSTR 0 (a,ARB)
                                  (λn. ind_type$BOTTOM)) a) ∨
                      (∃a. a0' =
                           (λa.
                                ind_type$CONSTR (SUC 0) (ARB,a)
                                  (λn. ind_type$BOTTOM)) a) ∨
                      (∃a0 a1.
                         a0' =
                         (λa0 a1.
                              ind_type$CONSTR (SUC (SUC 0)) (ARB,ARB)
                                (ind_type$FCONS a0
                                   (ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
                           a0 a1 ∧ $var$('polynom') a0 ∧
                         $var$('polynom') a1) ∨
                      (∃a0 a1.
                         a0' =
                         (λa0 a1.
                              ind_type$CONSTR (SUC (SUC (SUC 0))) (ARB,ARB)
                                (ind_type$FCONS a0
                                   (ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
                           a0 a1 ∧ $var$('polynom') a0 ∧
                         $var$('polynom') a1) ∨
                      (∃a. a0' =
                           (λa.
                                ind_type$CONSTR (SUC (SUC (SUC (SUC 0))))
                                  (ARB,ARB)
                                  (ind_type$FCONS a (λn. ind_type$BOTTOM)))
                             a ∧ $var$('polynom') a) ⇒
                      $var$('polynom') a0') ⇒
                   $var$('polynom') a0') rep
   
   [polynom_case_def]  Definition
      
      ⊢ (∀a f f1 f2 f3 f4. polynom_CASE (Pvar a) f f1 f2 f3 f4 = f a) ∧
        (∀a f f1 f2 f3 f4. polynom_CASE (Pconst a) f f1 f2 f3 f4 = f1 a) ∧
        (∀a0 a1 f f1 f2 f3 f4.
           polynom_CASE (Pplus a0 a1) f f1 f2 f3 f4 = f2 a0 a1) ∧
        (∀a0 a1 f f1 f2 f3 f4.
           polynom_CASE (Pmult a0 a1) f f1 f2 f3 f4 = f3 a0 a1) ∧
        ∀a f f1 f2 f3 f4. polynom_CASE (Popp a) f f1 f2 f3 f4 = f4 a
   
   [polynom_normalize_def]  Definition
      
      ⊢ (∀r i. polynom_normalize r (Pvar i) = Cons_varlist [i] Nil_monom) ∧
        (∀r c. polynom_normalize r (Pconst c) = Cons_monom c [] Nil_monom) ∧
        (∀r pl pr.
           polynom_normalize r (Pplus pl pr) =
           r_canonical_sum_merge r (polynom_normalize r pl)
             (polynom_normalize r pr)) ∧
        (∀r pl pr.
           polynom_normalize r (Pmult pl pr) =
           r_canonical_sum_prod r (polynom_normalize r pl)
             (polynom_normalize r pr)) ∧
        ∀r p.
          polynom_normalize r (Popp p) =
          r_canonical_sum_scalar3 r (r.RN r.R1) [] (polynom_normalize r p)
   
   [polynom_simplify_def]  Definition
      
      ⊢ ∀r x.
          polynom_simplify r x =
          r_canonical_sum_simplify r (polynom_normalize r x)
   
   [polynom_size_def]  Definition
      
      ⊢ (∀f a. polynom_size f (Pvar a) = 1 + index_size a) ∧
        (∀f a. polynom_size f (Pconst a) = 1 + f a) ∧
        (∀f a0 a1.
           polynom_size f (Pplus a0 a1) =
           1 + (polynom_size f a0 + polynom_size f a1)) ∧
        (∀f a0 a1.
           polynom_size f (Pmult a0 a1) =
           1 + (polynom_size f a0 + polynom_size f a1)) ∧
        ∀f a. polynom_size f (Popp a) = 1 + polynom_size f a
   
   [r_canonical_sum_merge_def]  Definition
      
      ⊢ ∀r. r_canonical_sum_merge r = canonical_sum_merge (semi_ring_of r)
   
   [r_canonical_sum_prod_def]  Definition
      
      ⊢ ∀r. r_canonical_sum_prod r = canonical_sum_prod (semi_ring_of r)
   
   [r_canonical_sum_scalar2_def]  Definition
      
      ⊢ ∀r. r_canonical_sum_scalar2 r =
            canonical_sum_scalar2 (semi_ring_of r)
   
   [r_canonical_sum_scalar3_def]  Definition
      
      ⊢ ∀r. r_canonical_sum_scalar3 r =
            canonical_sum_scalar3 (semi_ring_of r)
   
   [r_canonical_sum_scalar_def]  Definition
      
      ⊢ ∀r. r_canonical_sum_scalar r =
            canonical_sum_scalar (semi_ring_of r)
   
   [r_canonical_sum_simplify_def]  Definition
      
      ⊢ ∀r. r_canonical_sum_simplify r =
            canonical_sum_simplify (semi_ring_of r)
   
   [r_ics_aux_def]  Definition
      
      ⊢ ∀r. r_ics_aux r = ics_aux (semi_ring_of r)
   
   [r_interp_cs_def]  Definition
      
      ⊢ ∀r. r_interp_cs r = interp_cs (semi_ring_of r)
   
   [r_interp_m_def]  Definition
      
      ⊢ ∀r. r_interp_m r = interp_m (semi_ring_of r)
   
   [r_interp_sp_def]  Definition
      
      ⊢ ∀r. r_interp_sp r = interp_sp (semi_ring_of r)
   
   [r_interp_vl_def]  Definition
      
      ⊢ ∀r. r_interp_vl r = interp_vl (semi_ring_of r)
   
   [r_ivl_aux_def]  Definition
      
      ⊢ ∀r. r_ivl_aux r = ivl_aux (semi_ring_of r)
   
   [r_monom_insert_def]  Definition
      
      ⊢ ∀r. r_monom_insert r = monom_insert (semi_ring_of r)
   
   [r_spolynom_normalize_def]  Definition
      
      ⊢ ∀r. r_spolynom_normalize r = spolynom_normalize (semi_ring_of r)
   
   [r_spolynom_simplify_def]  Definition
      
      ⊢ ∀r. r_spolynom_simplify r = spolynom_simplify (semi_ring_of r)
   
   [r_varlist_insert_def]  Definition
      
      ⊢ ∀r. r_varlist_insert r = varlist_insert (semi_ring_of r)
   
   [canonical_sum_merge_def]  Theorem
      
      ⊢ ∀r. (∀t2 t1 l2 l1 c2 c1.
               r_canonical_sum_merge r (Cons_monom c1 l1 t1)
                 (Cons_monom c2 l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS =>
                   Cons_monom c1 l1
                     (r_canonical_sum_merge r t1 (Cons_monom c2 l2 t2))
               | EQUAL =>
                 Cons_monom (r.RP c1 c2) l1 (r_canonical_sum_merge r t1 t2)
               | GREATER =>
                 Cons_monom c2 l2
                   (r_canonical_sum_merge r (Cons_monom c1 l1 t1) t2)) ∧
            (∀t2 t1 l2 l1 c1.
               r_canonical_sum_merge r (Cons_monom c1 l1 t1)
                 (Cons_varlist l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS =>
                   Cons_monom c1 l1
                     (r_canonical_sum_merge r t1 (Cons_varlist l2 t2))
               | EQUAL =>
                 Cons_monom (r.RP c1 r.R1) l1
                   (r_canonical_sum_merge r t1 t2)
               | GREATER =>
                 Cons_varlist l2
                   (r_canonical_sum_merge r (Cons_monom c1 l1 t1) t2)) ∧
            (∀t2 t1 l2 l1 c2.
               r_canonical_sum_merge r (Cons_varlist l1 t1)
                 (Cons_monom c2 l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS =>
                   Cons_varlist l1
                     (r_canonical_sum_merge r t1 (Cons_monom c2 l2 t2))
               | EQUAL =>
                 Cons_monom (r.RP r.R1 c2) l1
                   (r_canonical_sum_merge r t1 t2)
               | GREATER =>
                 Cons_monom c2 l2
                   (r_canonical_sum_merge r (Cons_varlist l1 t1) t2)) ∧
            (∀t2 t1 l2 l1.
               r_canonical_sum_merge r (Cons_varlist l1 t1)
                 (Cons_varlist l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS =>
                   Cons_varlist l1
                     (r_canonical_sum_merge r t1 (Cons_varlist l2 t2))
               | EQUAL =>
                 Cons_monom (r.RP r.R1 r.R1) l1
                   (r_canonical_sum_merge r t1 t2)
               | GREATER =>
                 Cons_varlist l2
                   (r_canonical_sum_merge r (Cons_varlist l1 t1) t2)) ∧
            (∀s1. r_canonical_sum_merge r s1 Nil_monom = s1) ∧
            (∀v6 v5 v4.
               r_canonical_sum_merge r Nil_monom (Cons_monom v4 v5 v6) =
               Cons_monom v4 v5 v6) ∧
            ∀v8 v7.
              r_canonical_sum_merge r Nil_monom (Cons_varlist v7 v8) =
              Cons_varlist v7 v8
   
   [canonical_sum_prod_def]  Theorem
      
      ⊢ ∀r. (∀c1 l1 t1 s2.
               r_canonical_sum_prod r (Cons_monom c1 l1 t1) s2 =
               r_canonical_sum_merge r (r_canonical_sum_scalar3 r c1 l1 s2)
                 (r_canonical_sum_prod r t1 s2)) ∧
            (∀l1 t1 s2.
               r_canonical_sum_prod r (Cons_varlist l1 t1) s2 =
               r_canonical_sum_merge r (r_canonical_sum_scalar2 r l1 s2)
                 (r_canonical_sum_prod r t1 s2)) ∧
            ∀s2. r_canonical_sum_prod r Nil_monom s2 = Nil_monom
   
   [canonical_sum_scalar2_def]  Theorem
      
      ⊢ ∀r. (∀l0 c l t.
               r_canonical_sum_scalar2 r l0 (Cons_monom c l t) =
               r_monom_insert r c (list_merge index_lt l0 l)
                 (r_canonical_sum_scalar2 r l0 t)) ∧
            (∀l0 l t.
               r_canonical_sum_scalar2 r l0 (Cons_varlist l t) =
               r_varlist_insert r (list_merge index_lt l0 l)
                 (r_canonical_sum_scalar2 r l0 t)) ∧
            ∀l0. r_canonical_sum_scalar2 r l0 Nil_monom = Nil_monom
   
   [canonical_sum_scalar3_def]  Theorem
      
      ⊢ ∀r. (∀c0 l0 c l t.
               r_canonical_sum_scalar3 r c0 l0 (Cons_monom c l t) =
               r_monom_insert r (r.RM c0 c) (list_merge index_lt l0 l)
                 (r_canonical_sum_scalar3 r c0 l0 t)) ∧
            (∀c0 l0 l t.
               r_canonical_sum_scalar3 r c0 l0 (Cons_varlist l t) =
               r_monom_insert r c0 (list_merge index_lt l0 l)
                 (r_canonical_sum_scalar3 r c0 l0 t)) ∧
            ∀c0 l0. r_canonical_sum_scalar3 r c0 l0 Nil_monom = Nil_monom
   
   [canonical_sum_scalar_def]  Theorem
      
      ⊢ ∀r. (∀c0 c l t.
               r_canonical_sum_scalar r c0 (Cons_monom c l t) =
               Cons_monom (r.RM c0 c) l (r_canonical_sum_scalar r c0 t)) ∧
            (∀c0 l t.
               r_canonical_sum_scalar r c0 (Cons_varlist l t) =
               Cons_monom c0 l (r_canonical_sum_scalar r c0 t)) ∧
            ∀c0. r_canonical_sum_scalar r c0 Nil_monom = Nil_monom
   
   [canonical_sum_simplify_def]  Theorem
      
      ⊢ ∀r. (∀c l t.
               r_canonical_sum_simplify r (Cons_monom c l t) =
               if c = r.R0 then r_canonical_sum_simplify r t
               else if c = r.R1 then
                 Cons_varlist l (r_canonical_sum_simplify r t)
               else Cons_monom c l (r_canonical_sum_simplify r t)) ∧
            (∀l t.
               r_canonical_sum_simplify r (Cons_varlist l t) =
               Cons_varlist l (r_canonical_sum_simplify r t)) ∧
            r_canonical_sum_simplify r Nil_monom = Nil_monom
   
   [datatype_polynom]  Theorem
      
      ⊢ DATATYPE (polynom Pvar Pconst Pplus Pmult Popp)
   
   [ics_aux_def]  Theorem
      
      ⊢ ∀r. (∀vm a. r_ics_aux r vm a Nil_monom = a) ∧
            (∀vm a l t.
               r_ics_aux r vm a (Cons_varlist l t) =
               r.RP a (r_ics_aux r vm (r_interp_vl r vm l) t)) ∧
            ∀vm a c l t.
              r_ics_aux r vm a (Cons_monom c l t) =
              r.RP a (r_ics_aux r vm (r_interp_m r vm c l) t)
   
   [interp_cs_def]  Theorem
      
      ⊢ ∀r. (∀vm. r_interp_cs r vm Nil_monom = r.R0) ∧
            (∀vm l t.
               r_interp_cs r vm (Cons_varlist l t) =
               r_ics_aux r vm (r_interp_vl r vm l) t) ∧
            ∀vm c l t.
              r_interp_cs r vm (Cons_monom c l t) =
              r_ics_aux r vm (r_interp_m r vm c l) t
   
   [interp_m_def]  Theorem
      
      ⊢ ∀r. (∀vm c. r_interp_m r vm c [] = c) ∧
            ∀vm c x t.
              r_interp_m r vm c (x::t) = r.RM c (r_ivl_aux r vm x t)
   
   [interp_sp_def]  Theorem
      
      ⊢ ∀r. (∀vm c. r_interp_sp r vm (SPconst c) = c) ∧
            (∀vm i. r_interp_sp r vm (SPvar i) = varmap_find i vm) ∧
            (∀vm p1 p2.
               r_interp_sp r vm (SPplus p1 p2) =
               r.RP (r_interp_sp r vm p1) (r_interp_sp r vm p2)) ∧
            ∀vm p1 p2.
              r_interp_sp r vm (SPmult p1 p2) =
              r.RM (r_interp_sp r vm p1) (r_interp_sp r vm p2)
   
   [interp_vl_def]  Theorem
      
      ⊢ ∀r. (∀vm. r_interp_vl r vm [] = r.R1) ∧
            ∀vm x t. r_interp_vl r vm (x::t) = r_ivl_aux r vm x t
   
   [ivl_aux_def]  Theorem
      
      ⊢ ∀r. (∀vm x. r_ivl_aux r vm x [] = varmap_find x vm) ∧
            ∀vm x x' t'.
              r_ivl_aux r vm x (x'::t') =
              r.RM (varmap_find x vm) (r_ivl_aux r vm x' t')
   
   [monom_insert_def]  Theorem
      
      ⊢ ∀r. (∀t2 l2 l1 c2 c1.
               r_monom_insert r c1 l1 (Cons_monom c2 l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS => Cons_monom c1 l1 (Cons_monom c2 l2 t2)
               | EQUAL => Cons_monom (r.RP c1 c2) l1 t2
               | GREATER => Cons_monom c2 l2 (r_monom_insert r c1 l1 t2)) ∧
            (∀t2 l2 l1 c1.
               r_monom_insert r c1 l1 (Cons_varlist l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS => Cons_monom c1 l1 (Cons_varlist l2 t2)
               | EQUAL => Cons_monom (r.RP c1 r.R1) l1 t2
               | GREATER => Cons_varlist l2 (r_monom_insert r c1 l1 t2)) ∧
            ∀l1 c1.
              r_monom_insert r c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom
   
   [polynom_11]  Theorem
      
      ⊢ (∀a a'. Pvar a = Pvar a' ⇔ a = a') ∧
        (∀a a'. Pconst a = Pconst a' ⇔ a = a') ∧
        (∀a0 a1 a0' a1'. Pplus a0 a1 = Pplus a0' a1' ⇔ a0 = a0' ∧ a1 = a1') ∧
        (∀a0 a1 a0' a1'. Pmult a0 a1 = Pmult a0' a1' ⇔ a0 = a0' ∧ a1 = a1') ∧
        ∀a a'. Popp a = Popp a' ⇔ a = a'
   
   [polynom_Axiom]  Theorem
      
      ⊢ ∀f0 f1 f2 f3 f4. ∃fn.
          (∀a. fn (Pvar a) = f0 a) ∧ (∀a. fn (Pconst a) = f1 a) ∧
          (∀a0 a1. fn (Pplus a0 a1) = f2 a0 a1 (fn a0) (fn a1)) ∧
          (∀a0 a1. fn (Pmult a0 a1) = f3 a0 a1 (fn a0) (fn a1)) ∧
          ∀a. fn (Popp a) = f4 a (fn a)
   
   [polynom_case_cong]  Theorem
      
      ⊢ ∀M M' f f1 f2 f3 f4.
          M = M' ∧ (∀a. M' = Pvar a ⇒ f a = f' a) ∧
          (∀a. M' = Pconst a ⇒ f1 a = f1' a) ∧
          (∀a0 a1. M' = Pplus a0 a1 ⇒ f2 a0 a1 = f2' a0 a1) ∧
          (∀a0 a1. M' = Pmult a0 a1 ⇒ f3 a0 a1 = f3' a0 a1) ∧
          (∀a. M' = Popp a ⇒ f4 a = f4' a) ⇒
          polynom_CASE M f f1 f2 f3 f4 = polynom_CASE M' f' f1' f2' f3' f4'
   
   [polynom_case_eq]  Theorem
      
      ⊢ polynom_CASE x f f1 f2 f3 f4 = v ⇔
        (∃i. x = Pvar i ∧ f i = v) ∨ (∃a. x = Pconst a ∧ f1 a = v) ∨
        (∃p p0. x = Pplus p p0 ∧ f2 p p0 = v) ∨
        (∃p p0. x = Pmult p p0 ∧ f3 p p0 = v) ∨ ∃p. x = Popp p ∧ f4 p = v
   
   [polynom_distinct]  Theorem
      
      ⊢ (∀a' a. Pvar a ≠ Pconst a') ∧ (∀a1 a0 a. Pvar a ≠ Pplus a0 a1) ∧
        (∀a1 a0 a. Pvar a ≠ Pmult a0 a1) ∧ (∀a' a. Pvar a ≠ Popp a') ∧
        (∀a1 a0 a. Pconst a ≠ Pplus a0 a1) ∧
        (∀a1 a0 a. Pconst a ≠ Pmult a0 a1) ∧ (∀a' a. Pconst a ≠ Popp a') ∧
        (∀a1' a1 a0' a0. Pplus a0 a1 ≠ Pmult a0' a1') ∧
        (∀a1 a0 a. Pplus a0 a1 ≠ Popp a) ∧ ∀a1 a0 a. Pmult a0 a1 ≠ Popp a
   
   [polynom_induction]  Theorem
      
      ⊢ ∀P. (∀i. P (Pvar i)) ∧ (∀a. P (Pconst a)) ∧
            (∀p p0. P p ∧ P p0 ⇒ P (Pplus p p0)) ∧
            (∀p p0. P p ∧ P p0 ⇒ P (Pmult p p0)) ∧ (∀p. P p ⇒ P (Popp p)) ⇒
            ∀p. P p
   
   [polynom_nchotomy]  Theorem
      
      ⊢ ∀pp.
          (∃i. pp = Pvar i) ∨ (∃a. pp = Pconst a) ∨
          (∃p p0. pp = Pplus p p0) ∨ (∃p p0. pp = Pmult p p0) ∨
          ∃p. pp = Popp p
   
   [polynom_normalize_ok]  Theorem
      
      ⊢ ∀r. is_ring r ⇒
            ∀vm p.
              r_interp_cs r vm (polynom_normalize r p) = interp_p r vm p
   
   [polynom_simplify_ok]  Theorem
      
      ⊢ ∀r. is_ring r ⇒
            ∀vm p.
              r_interp_cs r vm (polynom_simplify r p) = interp_p r vm p
   
   [spolynom_normalize_def]  Theorem
      
      ⊢ ∀r. (∀i. r_spolynom_normalize r (SPvar i) =
                 Cons_varlist [i] Nil_monom) ∧
            (∀c. r_spolynom_normalize r (SPconst c) =
                 Cons_monom c [] Nil_monom) ∧
            (∀l r'.
               r_spolynom_normalize r (SPplus l r') =
               r_canonical_sum_merge r (r_spolynom_normalize r l)
                 (r_spolynom_normalize r r')) ∧
            ∀l r'.
              r_spolynom_normalize r (SPmult l r') =
              r_canonical_sum_prod r (r_spolynom_normalize r l)
                (r_spolynom_normalize r r')
   
   [spolynom_simplify_def]  Theorem
      
      ⊢ ∀r x.
          r_spolynom_simplify r x =
          r_canonical_sum_simplify r (r_spolynom_normalize r x)
   
   [varlist_insert_def]  Theorem
      
      ⊢ ∀r. (∀t2 l2 l1 c2.
               r_varlist_insert r l1 (Cons_monom c2 l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS => Cons_varlist l1 (Cons_monom c2 l2 t2)
               | EQUAL => Cons_monom (r.RP r.R1 c2) l1 t2
               | GREATER => Cons_monom c2 l2 (r_varlist_insert r l1 t2)) ∧
            (∀t2 l2 l1.
               r_varlist_insert r l1 (Cons_varlist l2 t2) =
               case list_compare index_compare l1 l2 of
                 LESS => Cons_varlist l1 (Cons_varlist l2 t2)
               | EQUAL => Cons_monom (r.RP r.R1 r.R1) l1 t2
               | GREATER => Cons_varlist l2 (r_varlist_insert r l1 t2)) ∧
            ∀l1.
              r_varlist_insert r l1 Nil_monom = Cons_varlist l1 Nil_monom
   
   
*)
end


Source File Identifier index Theory binding index

HOL 4, Kananaskis-14